Jan Haskovec1 Lisa Maria Kreusser2 Peter Markowich3
Abstract. Motivated by recent physics papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. The proof is based on reformulating the discrete energy functional as a sequence of integral functionals and proving their Γ-convergence towards the respective continuum energy functional.
Key words: Network formation; Γ-convergence; Continuum limit; Finite Element Discretization.
Math. Class. No.: 35K55; 92C42; 65M60
Contents
1. Introduction 2
2. An auxiliary Lemma 5
3. The 1D equidistant setting 6
3.1. Reformulation of the discrete energy functional 8
3.2. Convergence of the energy functionals 9
3.3. Introduction of diffusion and construction of continuum energy minimizers 11
4. The 2D rectangular equidistant setting 13
4.1. Finite element discretization of the Poisson equation 14
4.2. Reformulation of the discrete energy functional 16
4.3. Convergence of the energy functional 17
4.4. Introduction of diffusion and construction of continuum energy minimizers (γ >1) 19
5. Appendix 22
5.1. Linear basis functions 22
5.2. Gradients of ph 24
5.3. Explicit calculation for (4.8) 24
Acknowledgments 24
References 24
1Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Tech-nology, Thuwal 23955-6900, Kingdom of Saudi Arabia;[email protected]
2Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK;[email protected]
3
Mathematical and Computer Sciences and Engineering Division, King Abdullah University of Science and Technol-ogy, Thuwal 23955-6900, Kingdom of Saudi Arabia; Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria;[email protected]; [email protected]
Table 1. Notation. (*) denotes variables that are given as data.
Variable Meaning Related to
Sj (∗) intensity of source/sink vertex j∈V
Pj pressure vertex j∈V
Lij (∗) length of an edge edge (i, j)∈E Qij flow from j∈Vtoi∈V edge (i, j)∈E
Cij conductivity edge (i, j)∈E
1. Introduction
In this paper we derive the rigorous continuum limit of the discrete network formation model of Hu and Cai [13]. The model is posed on an a priori given graph G= (V,E), consisting of the set
of vertices (nodes) V and the set of unoriented edges (vessels) E. Any pair of vertices i, j ∈ V is
connected by at most one edge (i, j) ∈ E, such that the corresponding graph (V,E) is connected.
The lengths Lij >0 of the vessels (i, j)∈E are given a priori and fixed. The adjacency matrix of
the graph (V,E) is denoted by A, i.e., Aij = 1 if (i, j)∈E, otherwise Aij = 0.
Let us emphasize that by fixing (V,E), the set of possible flow directions in the network is also
fixed. For each node j ∈ V we prescribe the strength of source/sink Sj ∈ R and we adopt the
convention that Sj > 0 denotes sources, while Sj < 0 sinks. We also allow for Sj = 0, i.e., no external in- or outgoing flux in this node. We impose the global mass conservation
X
j∈V
Sj = 0. (1.1)
We denote Cij and, resp., Qij the conductivity and, resp., the flow through the vessel (i, j) ∈ E.
Note that the flow is oriented and we adopt the convention that Qij > 0 means net flow from node j ∈ V to node i ∈ V. An overview of the notation is provided in Table 1. We assume low
Reynolds number of the flow through the network, so that the flow rate through a vessel (i, j)∈E
is proportional to its conductivity and the pressure drop between its two ends, i.e.,
Qij =Cij
Pj−Pi
Lij
.
(1.2)
Local conservation of mass is expressed in terms of the Kirchhoff law,
X i∈V AijCij Pj−Pi Lij =Sj for allj ∈V. (1.3)
Note that for any given vector of conductivitiesC := (Cij)(i,j)∈E, (1.3) represents a linear system
of equations for the vector of pressures (Pj)j∈V. The system has a solution, unique up to an additive
constant, if and only if the graph with edge weights given by C is connected [7], where only edges with positive conductivities are taken into account (i.e., edges with zero conductivity are discarded). Assuming that the material cost for an edge (i, j)∈Eof the network is proportional to a power Cijγ of its conductivity, Hu and Cai [13] consider the energy consumption function of the form
E[C] := 1 2 X i∈V X j∈V Q2ij Cij +ν γC γ ij ! AijLij, (1.4)
whereν >0 is the metabolic coefficient andQij is given by (1.2), where the pressure drop Pj
−Pi
Lij is
(pumping power) of the material flow through the vessels, and we shall call it pumping term in the sequel. The second part represents the metabolic cost of maintaining the network and shall be called metabolic term. For instance, the metabolic cost for a blood vessel is proportional to its cross-section area [14]. Modeling blood flow by Hagen-Poiseuille’s law, the conductivity of the vessel is proportional to the square of its cross-section area. This implies γ = 1/2 for blood vessel systems. For leaf venations, the material cost is proportional to the number of small tubes, which is proportional toCij, and the metabolic cost is due to the effective loss of the photosynthetic power at the area of the venation cells, which is proportional to Cij1/2. Consequently, the effective value of γ typically used in models of leaf venation lies between 1/2 and 1, [13]. Hu and Cai showed that the optimal networks corresponding to minimizers of (1.3)-(1.4) exhibit a phase transition at
γ = 1, with a “uniform sheet” (the network is tiled with loops) for γ >1 and a “loopless tree” for
γ < 1, see also [11]. Moreover, they consider the gradient flow of the energy (1.4) constrained by the Kirchhoff law (1.3), which leads to the ODE system for the conductivities Cij,
dCij dt = Q2ij Cij2 −νC γ−1 ij ! Lij for (i, j)∈E,
coupled to the Kirchhoff law (1.3) through (1.2). This system represents an adaptation model which dynamically responds to local information and can naturally incorporate fluctuations in the flow.
This paper focuses on deriving the rigorous continuum limit of the energy functional (1.3)-(1.4) as the number of nodes of the underlying graph tends to infinity and the edge lengthsLij tend uniformly to zero. In a general setting with a sequence of unstructured graphs this is a mathematically very challenging task. In particular, one has to expect that the object obtained in the limit will depend on the structural and statistical properties of the graph sequence (connectivity, edge directions and density etc.). Therefore, we consider the particular setting where the graphs correspond to regular equidistant meshes in 1D and 2D. As we explain in Section 3, the energy minimization problem for (1.3)-(1.4) in the one-dimensional case is in fact trivial, and the form of the limiting functional is obvious. However, we use this setting as a toy example and carry out the rigorous limit passage anyway. The reason is that in the 1D setting we avoid most of the technical peculiarities of the two-dimensional case and we can focus on the essential idea of the method. Equipped with this insight, we shall turn to the two-dimensional case (Section 4), where the graph is an equidistant rectangular mesh on a square-shaped domain Ω.
In both the 1D and 2D cases, it is necessary to adopt the additional assumption that the con-ductivities are a priori bounded away from zero. In particular, we introduce a modification of the system (1.3)-(1.4) where the conductivities are of the form r+Cij, where r >0 is a fixed global constant. The reason is that we need to guarantee the solvability of the Poisson equation (1.9) below, which will be obtained in the continuum limit. Moreover, in the 2D case, the additive terms in the energy functional have to be scaled by the square of the edge length Lij. This is due to the fact that we are embedding the inherently one-dimensional edges of the graph into two spatial dimensions; see [8, Section 3.2] for details. Thus, we shall work with the energy functional
E[C] := 1 2 X i∈V X j∈V Q2ij r+Cij +ν γ(r+Cij) γ ! AijLdij, (1.5)
where d= 1, 2 is the space dimension, coupled to the (properly rescaled) Kirchhoff law
X i∈V Aij(r+Cij) Pj−Pi Lij =LjSj for all j∈V (1.6)
through Qij = (r+Cij) Pj −Pi Lij , (1.7)
where Lj are (abstract) weights that scale linearly with the mean edge length; see [8, Section 3.1] for details about the scaling in (1.6). The main benefit of this paper is the rigorous derivation of the limiting energy functional, which for the two-dimensional case is of the form
E[c] = Z Ω ∇p[c]·(rI+c)∇p[c] +ν γ (|r+c1| γ+|r+c 2|γ) dx, (1.8)
with x= (x, y)∈R2 and wherep[c]∈H1(Ω) is a weak solution of the Poisson equation −∇ ·((rI+c)∇p) =S
(1.9)
subject to no-flux boundary conditions on ∂Ω, where I is the unit matrix and c is the diagonal 2×2-tensor c= c1 0 0 c2 . (1.10)
Here,S ∈L2(Ω) denotes the source/sink term and in analogy to (1.1) we requireR
ΩSdx= 0. The derivation is based on three steps:
(1) Establish a connection between the discrete solutions of the Kirchhoff law (1.6) and weak solutions of the Poisson equation (1.9); see Section 3.1 in 1D and Sections 4.1, 4.2 in 2D. (2) Reformulate the discrete energy functional (1.5) as an integral functional defined on the set
of bounded functions; see Section 3.1 in 1D and Section 4.2 in 2D.
(3) Show that the sequence of integral functionals Γ-converges to the energy functional (1.8); see Section 3.2 in 1D and Section 4.3 in 2D. See, e.g., [6, 2] for details about Γ-convergence. The Γ-convergence opens the door for constructing global minimizers of (1.8)–(1.9) as limits of sequences of minimizers of the discrete problem (1.5)–(1.6). However, for this we need strong con-vergence of the minimizers in an appropriate topology. In agreement with [9, 10, 3] we introduce diffusive terms into the discrete energy functionals, modeling random fluctuations in the medium (Section 3.3 for 1D and Section 4.4 in 2D). The diffusive terms provide compactness of the min-imizing sequences in a suitable topology and facilitate the construction of global minimizers of (1.8)–(1.9).
Let us note that the steepest descent minimization procedure for (1.8)–(1.9) is represented by the formalL2-gradient flow. This leads to the system of partial differential equations forc1 =c1(t, x, y),
c2 =c2(t, x, y),
∂tc1 = (∂xp)2−ν(r+c1)γ−1,
∂tc2 = (∂yp)2−ν(r+c2)γ−1, (1.11)
subject to homogeneous Dirichlet boundary data and coupled to (1.9) through (1.10). The exis-tence of weak solutions and their properties are studied in [8]. Finally, let us remark that [12] proposed a different PDE system, derived from the discrete model [13] by certain phenomenological considerations (laws of porous medium flow, see [3] for details). The system consists of a para-bolic reaction-diffusion equation for the vector-valued conductivity field, constrained by a Poisson equation for the pressure, and was studied in the series of papers [9, 10, 1, 3]. However, a rigorous derivation of the model is still lacking; moreover, no explicit connection to the system (1.11) has been established so far.
2. An auxiliary Lemma
Lemma 1. Fixr >0, a bounded domainΩ⊂Rdwithd≥1, andS ∈L2(Ω). Let(cN)
N∈N⊂L
∞(Ω)
be a sequence of nonnegative, essentially bounded functions on Ω, such thatcN →c∈L2(Ω)in the norm topology of L2(Ω). Let (pN)N∈N⊂H1(Ω)be a sequence of zero-average weak solutions of the
Poisson equation
−∇ ·((r+cN)∇pN) =S
(2.1)
subject to homogeneous Neumann boundary conditions on ∂Ω. Then ∇pN converges to ∇p and
√
cN∇pN converges to √c∇p strongly in L2(Ω), where p is the zero-average weak solution of
−∇ ·((r+c)∇p) =S
(2.2)
subject to homogeneous Neumann boundary conditions on ∂Ω. In particular, we have
lim N→∞ Z Ω (r+cN)|∇pN|2dx= Z Ω (r+c)|∇p|2dx. (2.3)
Remark 1. Note that we do not assume that (cN)N∈N is uniformly bounded in L
∞(Ω), nor that
c∈L∞(Ω).
Proof. UsingpN as a test function in (2.1), due to the nonnegativity ofcN, we have
r∇pN 2 L2(Ω)≤ Z Ω (r+cN)|∇pN|2dx = Z Ω SpNdx (2.4) ≤ 1 2εkSk 2 L2(Ω)+ εCP 2 ∇pN 2 L2(Ω),
where CP is the Poincar´e constant. With a suitable choice of ε >0 we obtain a uniform estimate on pN inH1(Ω). Consequently, there exists a subsequence of pN that converges weakly in H1(Ω) to somep∈H1(Ω). SincecN →c strongly inL2(Ω), we can pass to the limit in the distributional formulation of (2.1) to obtain Z Ω (r+c)∇p· ∇φdx= Z Ω Sφdx for allφ∈C0∞(Ω). (2.5)
Noting that (2.4) also implies a uniform bound onR
Ωc
N|∇pN|2dx, we have due to the weak lower semicontinuity of theL2-norm,
Z Ω (r+c)|∇p|2dx≤lim inf N→∞ Z Ω (r+cN)|∇pN|2dx<+∞. (2.6)
Consequently, we can usep as a test function in (2.5) to obtain
Z Ω (r+c)|∇p|2dx= Z Ω Spdx.
Therefore, using pN as a test function in (2.1), lim N→∞ Z Ω (r+cN)|∇pN|2dx= lim N→∞ Z Ω SpNdx= Z Ω Spdx= Z Ω (r+c)|∇p|2dx, which gives (2.3) and, further,
lim sup N→∞ Z Ω |∇pN|2dx ≤ lim sup N→∞ Z Ω (r+cN)|∇pN|2dx+ lim sup N→∞ − Z Ω cN|∇pN|2dx = Z Ω (r+c)|∇p|2dx−lim inf N→∞ Z Ω cN|∇pN|2dx.
Now, using (2.6), we have −lim inf N→∞ Z Ω cN|∇pN|2dx=−lim inf N→∞ Z Ω |√cN∇pN|2dx≤ − Z Ω |√c∇p|2dx. Therefore, lim sup N→∞ Z Ω |∇pN|2dx≤ Z Ω |∇p|2dx, so that limN→∞ ∇pN
L2(Ω) =k∇pkL2(Ω), which directly implies that (a subsequence of) pN
con-verges towards pstrongly in H1(Ω).
3. The 1D equidistant setting
In this section we consider the spatially one-dimensional setting of the discrete network formation problem, where the graph (V,E) is given as a mesh on the interval [0,1]. Moreover, for simplicity
we consider the equidistant case, where for a fixed N ∈Nconstruct the sequence of meshpointsxi, xi =ih fori= 0, . . . , N, with h:= 1/N.
We identify the meshpoints xi with the vertices of the graph, i.e., we setV:= {xi; i= 0, . . . , N}.
The segments (xi−1, xi) connecting any two neighboring nodes are identified with the edges of the graph, i.e., E := {(xi−1, xi); i = 1, . . . , N}. By a slight abuse of notation, we shall write i ∈ V
instead ofxi∈Vin the sequel, and similarlyi∈Einstead of (xi−1, xi)∈E. Moreover, we shall use
the notation C := (Ci)Ni=1 withCi ≥0 the conductivity of the edge i∈E,Pi ∈Rfor the pressure
in nodei∈Vand SNi ∈Rfor the source/sink in nodei∈VwithPN
i=1SiN = 0 by (1.1). With this notation we rewrite the energy functional (1.5) as EN[C] :RN+ 7→R,
EN[C] :=h N X i=1 Q2i r+Ci + ν γ(r+Ci) γ, (3.1)
with the fluxes
Qi:= (r+Ci)
Pi−1−Pi
h , fori= 1, . . . , N.
(3.2)
Note that we orient the fluxes Qi such that Qi > 0 if the material flows from xi−1 to xi. The Kirchhoff law (1.6) is then written in the form
(r+Ci)Pi−Pi−1 h + (r+Ci+1) Pi−Pi+1 h =hS N i fori= 1, . . . , N−1, (3.3)
while for the terminal nodes we have (r+C1) P0−P1 h =hS N 0 , (r+CN) PN −PN−1 h =hS N N .
Obviously, in the 1D setting the fluxesQiare explicitly calculable from the given set of sources/sinks (Si)Ni=0 since the Kirchhoff law (3.3) is the chain of equations
Q1 = hS0N,
−Qi+Qi+1 = hSiN fori= 1, . . . , N−1,
which has the explicit solution Qi=h i−1 X j=0 SjN fori= 1, . . . , N−1. (3.4)
Note that due to the assumption of the global mass balance (1.1) the “terminal condition” fori=N
is implicitly satisfied, −QN =−h N−1 X j=0 Sj =hSNN. (3.5)
With the fluxes given by (3.4)–(3.5), it is trivial to find the global energy minimizer of (3.1), namely, (r+Ci)γ+1 =Q2i/ν. It is also easy to prove that the sequence of the functionals (3.1) converges as
h= 1/N →0 to the continuous functional
E[c] := Z 1 0 q(x)2 r+c(x) + ν γ(r+c(x)) γdx, (3.6)
with q(x) := R0xS(σ) dσ, in the sense of Riemannian sums if c is a continuous, nonnegative func-tion. Therefore, the limit passage to continuum description in the one-dimensional case is trivial. However, we shall use it as a “training example” which avoids most of the technical difficulties of the two-dimensional setting to gain a clear understanding of the main ideas of the method.
Therefore, we shall ignore the explicit formula (3.4) for the fluxes Qi and study the limit as
h= 1/N →0 of the sequence of energy functionals (3.1)–(3.2), i.e.,
EN[C] =h N X i=1 (r+Ci) Pi−Pi−1 h 2 +ν γ(r+Ci) γ, (3.7)
where the pressures Pi are calculated as a solution of the Kirchhoff law (3.3). Note that since
r +Ci > 0 for all i ∈ V, (3.3) is solvable, uniquely up to an additive constant. In the following
we shall show that the sequence (3.7) converges, as h = N1 → 0, to the functional (3.6) with
q := (r+c)∂xp[c], i.e., E[c] = Z 1 0 (r+c)(∂xp[c])2+ ν γ(r+c) γdx, (3.8)
where p[c]∈H1(0,1) is a weak solution of the Poisson equation
−∂x((r+c)∂xp) =S (3.9)
on (0,1), subject to no-flux boundary conditions. Here and in the sequel we fix the source/sink termS ∈L2(0,1) and, in agreement with (1.1), we assume the global mass balanceR01S(x) dx= 0. Since forc(x)≥0 the weak solutionp=p(x) of (3.9) is unique up to an additive constant, we shall, without loss of generality, always choose the zero-average solution, i.e., R01p(x) dx= 0.
We shall proceed in several steps: First, we put the discrete energy functionals (3.7) into an integral form, and find an equivalence between solutions of the Kirchhoff law (3.3) and the above Poisson equation with appropriate conductivity. Then we show the convergence of the sequence of reformulated discrete energy functionals towards a continuum one as h = 1/N → 0. Finally, we introduce a diffusive term into the energy functional, which will allow us to construct global minimizers of the continuum energy functional.
3.1. Reformulation of the discrete energy functional. In the first step we reformulate the energy functionals (3.7) such that they are defined on the space L∞+(0,1) of essentially bounded nonnegative functions on (0,1). For this purpose, we define the sequence of operatorsQN0 :RN →
L∞(0,1) by
QN0 : (Ci)Ni=1 7→c, withc(x)≡Ci forx∈(xi−1, xi), i= 1, . . . , N.
I.e., QN0 maps the sequence (Ci)iN=1 onto the bounded functionc =c(x), constant on each interval (xi−1, xi),i= 1, . . . , N. Then, we define the functionals EN :L∞+(0,1)7→R,
EN[c] := Z 1 0 (r+c) QN0 [∆hP] 2 + ν γ(r+c) γdx, (3.10) with (∆hP)i := Pi−Pi−1 h , i= 1, . . . , N, (3.11)
and P = (Pi)Ni=0 a solution of the Kirchhoff law (3.3) with the conductivities C= (Ci)Ni=1,
Ci:= 1 h Z xi xi−1 c(x) dx, i= 1, . . . , N.
Then, noting that for each C= (Ci)Ni=1 ∈RN +, 1 h Z xi xi−1 QN0 [C](x) dx=Ci for alli= 1, . . . , N,
the discrete energy functional (3.7) can be written in the integral form as EN[C] =EN[
QN0 [C]]. Moreover, we establish a connection between the solutions of the Kirchhoff law (3.3) and weak solutions of the Poisson equation (3.9) with c=QN0 [C]:
Lemma 2. For any C = (Ci)Ni=1 ∈ RN+ and S ∈ L2(0,1) with
R1
0 S(x) dx = 0, let p = p(x) ∈
H1(0,1) be a weak solution of the Poisson equation (3.9) withc=QN0 [C], i.e.,
−∂x (r+QN0 [C])∂xp
=S,
(3.12)
subject to no-flux boundary conditions on (0,1). Then,
Pi :=p(xi), i= 0, . . . , N, (3.13)
is a solution of the Kirchhoff law (3.3) with the conductivities C = (Ci)Ni=1 and the source/sink terms SiN := 1 h Z 1 0 S(x)φNi (x) dx, i= 0, . . . , N, (3.14)
with the hat functions φN
i =φNi (x) defined in (3.15) below.
Proof. Note that for any C ∈ RN
+ there exists a weak solution p = p(x) ∈ H1(0,1) of (3.12), unique up to an additive constant. Fori= 1, . . . , N we construct the family of piecewise linear test functions φNi , supported on (xi−1, xi+1), with
φNi (x) = ( 1 +x−xi h forx∈(xi−1, xi), 1−x−xi h forx∈(xi, xi+1). . (3.15)
Using the hat function φNi as a test function in (3.12), we obtain (r+Ci) p(xi)−p(xi−1) h + (r+Ci+1) p(xi)−p(xi+1) h =hS N i ,
where we used the fact that, by construction, QN0 [C]≡Ci on the interval (xi−1, xi). Note that due to the embeddingH1(0,1),→C(0,1) any weak solutionp=p(x) of (3.12) is a continuous function on [0,1], so the pointwise values p(xi) are well defined for all i = 0, . . . , N. Thus, defining Pi as in (3.13) we obtain a solution of the Kirchhoff law (3.3) with the conductivities C = (Ci)Ni=1 and source/sink terms (3.14).
Note that since h1R1
0 φ N
i (x) dx= 1 and S ∈L2(0,1), the Lebesgue differentiation theorem gives
SiN = 1
h
Z 1
0
S(x)φNi (x) dx→S(x) for a.e. x=xi ash= 1/N →0.
Consequently, for a fixedS∈L2(0,1) and anyN ∈N, we have the following reformulation of the
discrete problem:
Proposition 1. For any vector C = (Ci)Ni=1∈RN+, we have
EN[C] =EN[QN0 [C]],
whereEN[C]is the discrete energy functional (3.7)coupled to the Kirchhoff law (3.3)with sources/sinks
SN
i given by (3.14), andEN is the integral form (3.10)–(3.11)with the pressures given byPi =p(xi),
i= 0, . . . , N, where p∈H1(0,1) solves the Poisson equation (3.12).
3.2. Convergence of the energy functionals. Due to Proposition 1, we are motivated to prove the convergence of the sequence of functionalsEN given by (3.10)–(3.11) towardsE[c] given by (3.8) with p[c] ∈ H1(0,1) a weak solution of (3.9) with conductivity c = c(x), equipped with no-flux boundary conditions. We choose to work in the space of essentially bounded functions on (0,1) equipped with the topology of L2(0,1). The choice of topology is motivated by the need for strong convergence of piecewise constant approximations of bounded functions. Of course, this is true in
Lq(0,1) with any q <+∞; our particular choice of L2(0,1) is further dictated by the fact that we shall apply Lemma 1 in the sequel.
Lemma 3. Let γ ≥0. For any sequence of nonnegative functions (cN)N∈N, uniformly bounded in L∞(0,1)and such that cN →c in the norm topology of L2(0,1) asN → ∞, we have
EN[cN]→ E[c] ash= 1/N →0.
Proof. By assumption, cN → c in the norm topology of L2(0,1). Consequently, there is a sub-sequence converging almost everywhere on (0,1) to c, and thus r+cN(x)γ converges almost everywhere to (r+c(x))γ. Since, by assumption, the sequence r+cN(x)γ is uniformly bounded inL∞(0,1), we have by the dominated convergence theorem
Z 1 0 r+cN(x)γ dx→ Z 1 0 (r+c(x))γdx as h= 1/N →0.
We recall that the pumping part of the discrete energy EN[cN] (3.10) is of the form
Z 1 0 (r+cN) QN0 [∆hpN] 2 dx, (3.16) with (∆hpN)i := pN(xi)−pN(xi−1) h , i= 1, . . . , N,
wherepN ∈H1(0,1) is a solution of the Poisson equation (3.9) with conductivitycN, subject to the no-flux boundary condition. Let us show that (a subsequence of) QN0 [∆hpN] converges to ∂xp[c] strongly in L2(0,1). We proceed in three steps:
• Weak convergence. By Jensen inequality we have
Q N 0 [∆hpN] 2 L2(0,1) = h N X i=1 pN(xi)−pN(xi−1) h 2 (3.17) = h N X i=1 1 h Z xi xi−1 ∂xpN(x) dx !2 ≤ Z 1 0 (∂xpN)2dx.
Due to the nonnegativity of the functions cN, the right-hand side is uniformly bounded. Consequently, there exists a weakly converging subsequence of QN0 [∆pN] inL2(0,1).
• Identification of the limit. For a smooth, compactly supported test function ψ ∈ C0∞(0,1) we write Z 1 0 QN0 [∆hpN](x)ψ(x) dx = N X i=1 pN(xi)−pN(xi−1) h Z xi xi−1 ψ(x) dx = 1 h N−1 X i=1 pN(xi) Z xi xi−1 ψ(x) dx− Z xi+1 xi ψ(x) dx ! + “boundary terms”,
where “boundary terms” are the two terms with i= 0 and i=N, which we however can neglect for large enough N since ψ has a compact support. Then, Taylor expansion for ψ
gives Z xi xi−1 ψ(x) dx− Z xi+1 xi ψ(x) dx=−h Z xi xi−1 ∂xψ(x) dx+ h2 2 Z xi xi−1 ∂xx2 ψ(ξ(x)) dx,
withξ(x)∈(xi−1, xi). Due to the estimate
h2 2 Z xi xi−1 ∂2xxψ(ξ(x)) dx ≤ h 3 2 ∂2xxψ L∞(0,1) we have Z xi xi−1 ψ(x) dx− Z xi+1 xi ψ(x) dx=−h Z xi xi−1 ∂xψ(x) dx+O(h3), so that Z 1 0 QN0 [∆hpN](x)ψ(x) dx = − Z 1 0 pN∂ xψ(x) dx+O(h), wherepN is the piecewise constant function
pN(x)≡pN(x
i) forx∈(xi−1, xi], i= 1, . . . , N.
It is easy to check that, due to the strong convergence of cN towards c in L2(0,1), pN
(a subsequence of) pN converges uniformly to p[c] on (0,1), and, therefore pN converges strongly top[c]. Therefore, Z 1 0 QN0 [∆hpN](x)ψ(x) dx → − Z 1 0 p(x)∂xψ(x) dx ash= 1/N →0, = Z 1 0 ψ(x)∂xp(x) dx.
We conclude that weak limit of (the subsequence of) QN0 [∆hpN] is∂xp[c].
• Strong convergence. Finally, due to (3.17), we have
Q N 0 [∆hpN]−∂xp[c] 2 L2(0,1) = Q N 0 [∆hpN] 2 L2(0,1)−2hQ N 0 [∆hpN], ∂xp[c]iL2(0,1)+k∂xp[c]k2L2(0,1) ≤ ∂xpN 2 L2(0,1)−2hQ N 0 [∆hpN], ∂xp[c]iL2(0,1)+k∂xp[c]k2L2(0,1),
which vanishes in the limit h = 1/N → 0 due to the weak convergence of QN0 [∆pN] and strong convergence of∂xpN inL2(0,1) due to Lemma 1. Thus,QN0 [∆pN] converges strongly to∂xp[c] inL2(0,1).
We conclude that due to the weak-∗convergence of (r+cN) towards (r+c) inL∞(0,1), and strong convergence of QN0 [∆hpN]
2
towards (∂xp[c])2inL1(0,1), we can pass to the limit ash= 1/N →0 in (3.16) to obtain
Z 1
0
(r+c) (∂xp[c])2 dx.
3.3. Introduction of diffusion and construction of continuum energy minimizers. In Lemma 3 we proved the convergence of the sequence of energy functionals EN towards E, i.e., for any cN → c in the norm topology of L2(0,1), we have EN[cN] → E[c] as N → ∞. In order to construct energy minimizers of E as limits of sequences of minimizers of the functionals EN, we need to introduce a term into EN that shall guarantee compactness of the sequence of discrete minimizers. This is done, in agreement with [9, 10, 3], by introducing a diffusive term into the discrete energy functional (3.7), modeling random fluctuations in the medium. Thus, we construct the sequenceEdiffN :RN+ →R,
EdiffN [C] :=D2h N−1 X i=1 Ci+1−Ci h 2 +EN[C], (3.18)
with EN[C] defined in (3.7), coupled to the Kirchhoff law (3.3) with sources/sinks SiN given by (3.14), and D2 > 0 the diffusion constant. Note that the new term is a discrete Laplacian acting on the conductivitiesC.
We now need to reformulate the discrete energy functionals (3.18) in terms of integrals. For this sake, we construct the sequence of operators QN1 : RN → C(0,1), where QN1 [C] is a continuous function on [0,1], linear on each interval (xi−h/2, xi+h/2), with
QN1 [C](xi−h/2) =Ci fori= 1, . . . , N, and
Then we write the finite difference term in (3.18) as D2h N−1 X i=1 Ci+1−Ci h 2 =D2 Z 1 0 ∂xQN1 [C] 2 dx, and we have
Proposition 2. For any vector C = (Ci)Ni=1∈RN +, EdiffN [C] =D2 Z 1 0 ∂xQN1 [C] 2 dx+EN[QN0 [C]],
whereEdiffN defined in (3.18)andEN is given by (3.10)–(3.11)with the pressures given byP
i =p(xi),
i= 0, . . . , N, where p∈H1(0,1) solves the Poisson equation (3.12).
We are now ready to prove the main result of this section:
Theorem 1. Let γ ≥ 0, S ∈ L2(0,1) with R01S(x) dx and SiN given by (3.14). Let (CN)N∈N be
a sequence of global minimizers of the discrete energy functionals EdiffN given by (3.18). Then the sequenceQN1 [CN]converges weakly inH1(0,1)toc∈H1(0,1), a global minimizer of the functional
Ediff :H1 +(0,1)→R, Ediff[c] :=D2 Z 1 0 (∂xc)2 dx+E[c], where E[c] is given by (3.8). Proof. Let us observe that
EdiffN [CN]≤EdiffN [0] =rh N X i=1 e Pi−Pei−1 h !2 + ν γr γ,
where (Pei)Ni=1 is a solution of the Kirchhoff law (3.3) with zero conductivities and sources/sinks
given by (3.14). Thus, Pei =pe(xi) for i= 1, . . . , N, wherepe=pe(x) is a weak solution of−r∆p=S
subject to no-flux boundary conditions. Then we have by the Jensen inequality
D2h N X i=1 e Pi−Pei−1 h !2 = D2h N X i=1 1 h Z xi xi−1 ∂xpedx !2 ≤ D2 Z 1 0 (∂xep) 2dx.
Consequently, the sequence EN
diff[CN] is uniformly bounded. Since the sequence
D2 Z 1 0 ∂xQN1 [CN] 2 dx=D2h N−1 X i=1 Ci+1−Ci h 2 ≤EdiffN [CN]
is uniformly bounded, there exists a subsequence ofQN1 [CN] converging to somec∈H1(0,1) weakly in H1(0,1), and strongly in L2(0,1); moreover, the sequence is uniformly bounded in L∞(0,1). It is easy to check that also QN0 [CN] converges to c strongly in L2(0,1), and is uniformly bounded
in L∞(0,1). Therefore, by Lemma 3, we have EN[CN] = EN[
QN0 [CN]] → E[c] as h = 1/N → 0. Moreover, due to the weak lower semicontinuity of the L2-norm, we have
Z 1 0 (∂xc)2 dx≤lim inf N→∞ Z 1 0 ∂xQN1 [CN] 2 dx . Consequently,
Ediff[c]≤lim inf N→∞ E
N diff[CN]. (3.19)
We claim that c is a global minimizer of Ediff in H+1(0,1). For contradiction, assume that there existsc∈H+1(0,1) such that
Ediff[c]<Ediff[c].
We define the sequence (CN)N∈N by
CNi := 1
h
Z xi
xi−1
c(x) dx, i= 1, . . . , N.
Then, by assumption, we have for all N ∈N,
EdiffN [CN]≥EdiffN [CN].
(3.20)
It is easy to check that the sequence QN1 [C N
] converges strongly in H1(0,1) towardsc, therefore
Z 1 0 ∂xQN1 [C N ] 2 dx→ Z 1 0 (∂xc)2 dx ash= 1/N →0. Moreover, the sequenceQN0 [C
N
] converges tocstrongly inL2(0,1), therefore, by Lemma 3,EN[
QN0 [C N ]]→ E[c] ash= 1/N →0. Consequently, lim h=1/N→0E N diff[C N ] =Ediff[c]<Ediff[c], a contradiction to (3.19)–(3.20).
4. The 2D rectangular equidistant setting
In this section we consider the spatially two-dimensional setting of the discrete network formation problem, where the graph (V,E) is embedded in the rectangle Ω := [0,1]2. We introduce the notation
x:= (x, y)∈Ω. ForN ∈Nwe construct the sequence of equidistant rectangular meshes in Ω with
mesh size h:= 1/N and mesh nodes Xi= (Xi, Yi),
Xi= (imod N+ 1)h, Yi = (idiv N + 1)h, fori= 0, . . . ,(N + 1)2−1, withh:= 1/N, where (idivN + 1) denotes the integer part of i/(N + 1) and (imod N + 1) the remainder. We identify the mesh nodes Xi = (Xi, Yi) with the vertices of the graph, i.e., we set V := {Xi; i = 0, . . . ,(N + 1)2 −1}. By a slight abuse of notation, we shall write i ∈ V instead of Xi ∈ V in
the sequel. For each node Xi, we denote by Xi,E, Xi,W, Xi,N, Xi,S its direct neighbors to the East, West, North and South, respectively (if they exist); see Fig. 1. Then, the set E of edges of
the graph is composed of the horizontal and vertical segments connecting the neighboring nodes, i.e., (Xi,Xi,?) for ? ∈ {E, W, N, S} and i ∈ V. We shall denote Ci? the conductivity of the edge (Xi,Xi,?), andPi, resp.,Pi,?, denote the pressure in the vertexXi, resp.,Xi,?. Similarly,Sihdenotes the source/sink in vertex i∈V.
Xi Xi,E Xi,W Xi,S Xi,N TN E i TSE i TS i TSW i TN W i TN i h h
Figure 1. Interior nodeXi with its four neighboring nodesXi,E,Xi,W,Xi,N,Xi,S and six adjacent triangles, TiN E,TiN,TiN W,TiSW,TiS,TiSE.
With this notation, the discrete energy functional (1.5) takes the particular form
Eh[C] = h 2 2 X i∈V X ?∈{E,W,N,S} (r+Ci?) Pi−Pi,? h 2 +ν γ(r+C ? i)γ, (4.1)
and the Kirchhoff law (1.6) is written as
X ?∈{E,W,N,S} (r+Ci?)Pi−Pi,? h =hS h i. fori∈V, (4.2)
For reasons explained later, we shall restrict to the case γ >1 in the sequel.
Our strategy is to perform a program analogous to the 1D case of Section 3: first, to put the discrete energy functionals (4.1) into an integral form and find an equivalence between solutions of the Kirchhoff law (4.2) and the above Poisson equation with appropriate conductivity. However, in the 2D case the situation is more complicated and we need to introduce a finite element discretization of the Poisson equation. We then establish a connection between the FE-discretization and the Kirchhoff law (4.2). In the next step we show the convergence of the sequence of reformulated discrete energy functionals towards a continuum one ash= 1/N →0, using standard results of the theory of finite elements. Finally, we introduce a diffusive term into the energy functional, which will allow us to construct global minimizers of the continuum energy functional.
4.1. Finite element discretization of the Poisson equation. We construct a regular trian-gulation on the domain Ω such that each interior node Xi has six adjacent triangles, TiN E, TiN,
TiN W,TiSW,TiS, TiSE, see Fig. 1. Boundary nodes have three, two or only one adjacent triangles, depending on their location. The union of the triangles adjacent to each Xi is denoted byUi. The collection of all triangles constructed in Ω is denoted by Th.
We fix S∈L2(Ω) withRΩSdx= 0 and consider a discretization of the Poisson equation
−∇ ·((rI+c)∇p) =S
(4.3)
on Ω subject to the no-flux boundary conditions, using the first-order (piecewise linear) H1 finite element method on the triangulation Th. Therefore, on each N E-triangle TN E
i we construct the linear basis functions φN Ei;1 ,φN Ei;2 ,φN Ei;3 with
φN Ei;1 (Xi) = 1, φN Ei;1 (Xi,E) = 0, φN Ei;1 (Xi,N) = 0,
φN Ei;2 (Xi) = 0, φN Ei;2 (Xi,E) = 1, φN Ei;2 (Xi,N) = 0,
φN Ei;3 (Xi) = 0, φN Ei;3 (Xi,E) = 0, φN Ei;3 (Xi,N) = 1,
and analogously for the other triangles in Ui, see Section 5.1 of the Appendix for explicit formulae. Denoting Wh ⊂H1(Ω) the space of continuous, piecewise linear functions on the triangulationTh, the finite element discretization of (4.3) reads
Z Ω ∇ph·(rI+c)∇ψhdx= Z Ω Sψhdx for allψh∈Wh. (4.4)
Using standard arguments (coercivity and continuity of the corresponding bilinear form) we con-struct a solution ph ∈ Wh of (4.4), unique up to an additive constant; without loss of generality we fix R
Ωp
h(x) dx = 0. The solution is represented by its vertex values Ph
i := ph(Xi), i ∈ V. In
particular, on each N E-triangleTiN E we have
ph(x) =PihφiN E;1 (x) +Pi,Eh φN Ei;2 (x) +Pi,Nh φN Ei;3 (x), x∈TiN E,
and the gradient of ph on TiN E is the constant vector
∇ph(x) = 1
h(P
h
i,E−Pih, Pi,Nh −Pih), x∈TiN E. (4.5)
Analogous formulae hold for all other triangles in Ui, as explicitly listed in Section 5.2 of the Appendix.
We now establish a connection between the discretized Poisson equation (4.4) and the Kirch-hoff law (4.2). For this purpose, we define the sequence of operators Qh0 mapping the vector of conductivities (Ci)i∈E onto piecewise constant 2×2 diagonal tensors,
Qh0 : (Ci)i∈E7→ c1 0 0 c2 . (4.6)
The functions c1 = c1(x), c2 = c2(x), defined on Ω, are constant on each triangle T ∈ Th and
c1 takes the value of the conductivity of the horizontal edge of T and c2 takes the value of the conductivity of the vertical edge ofT. In particular, we have
c1 := CiE onTiN E, CiE onTiSE, CiW on TiSW, CiW on TiN W, c2 := CiN onTiN E, CiS onTiS, CiS onTiSW, CiN onTiN. (4.7)
Then, for a given vector of conductivities C= (Ci)i∈E we consider the discretized Poisson equation
(4.4) with the conductivity tensor c:=Qh0[C]. For eachi∈Vwe construct the test functionψih as
with the basis functions on the right-hand side defined in Section 5.1 of the Appendix. Consequently, each ψhi is supported on Ui, linear on each triangle belonging to Ui, and continuous on Ω. Then, obviously, ψhi ∈Wh and using it as a test function in (4.4), we calculate, for the triangle TiN E,
Z TN E i ∇ph·rI+ Qh0[C] ∇ψh i dx= r+CiE 2 Pih−Pi,Eh +r+C N i 2 Pih−Pi,Nh ,
where we used (4.5), the identity ∇ψh
i ≡ −h1(1,1) on TiN E, and orthogonality relations between gradients of the basis functions (for instance,∇φN E
i;2 ·∇φN Ei;3 = 0). Performing analogous calculations for the remaining triangles constituting Ui, namely, TiSE, TiS,TiSW, TiN W and TiN, see Section 5.3 of the Appendix for explicit details, we obtain
Z Ω ∇ph·rI+ Qh0[C] ∇ψh i dx = X ?∈{E,W,N,S} (r+Ci?)(Pih−Pi,?h ). (4.8)
Consequently, (4.4) gives the identity
X ?∈{E,W,N,S} (r+Ci?)P h i −Pi,?h h = 1 h Z Ω Sψhi dx for all i∈V. Thus, defining
Sih:= 1 h2 Z Ω Sψihdx, (4.9)
we have the following result:
Lemma 4. For any vector of nonnegative conductivitiesC = (Ci)i∈E andS ∈L2(Ω)with
R
ΩSdx= 0, let ph ∈ Wh be a solution of the finite element discretization (4.4) with c := Qh0[C]. Then,
Ph
i :=ph(Xi),i∈V, is a solution of the rescaled Kirchhoff law (4.2)with the source/sink terms Sih
given by (4.9).
Note that since h12
R
Ωψhi(x) dx= 1, and, by assumption,S ∈L2(Ω), the Lebesgue differentiation theorem gives Sih= 1 h2 Z Ω Sψhi dx→S(x) for a.e. x=Xi as h= 1/N →0. Consequently, (Sih)h>0 is an approximating sequence for the datum S=S(x).
4.2. Reformulation of the discrete energy functional. We reformulate the energy functionals (4.1)–(4.2) such that they are defined on the space L∞+(Ω)2diag×2 of essentially bounded diagonal nonnegative tensors on Ω. We define the functional Eh:L∞
+(Ω)2 ×2 diag→R, Eh[c] := Z Ω ∇ph[c]·(rI+c)∇ph[c] +ν γ (|r+c1| γ+|r+c 2|γ) dx, (4.10)
where ph[c]∈Wh is a solution of the finite element problem (4.4).
Proposition 3. Let S ∈L2(Ω) withRΩSdx= 0 and Sih be given by (4.9). Then for any vector of nonnegative conductivities C= (Ci)i∈EN, we have
Eh[C] =Eh[Qh0[C]], with Eh defined in (4.1) and Eh given by (4.10).
Proof. We have shown in Section 4.1 that if ph = ph(x) denotes a solution of the finite element problem (4.4) withc=Qh0[C], then the vertex values Pih :=ph(Xi) satisfy the Kirchhoff law (4.2). Moreover, using (4.5) and the definition (4.6)–(4.7) of Qh0[C], we calculate
Z TN E i ∇ph·(rI+Qh0[C])∇phdx=|TiN E| (r+CiE) Pi,Eh −Pih h !2 + (r+CiN) P h i,N −Pih h !2
for each i ∈ V, and analogously for all other triangles. Noting that |TN E
i | = h2/2 and summing over all triangles, we obtain the formula (4.1) for the discrete energy Eh[C].
4.3. Convergence of the energy functional. With Proposition 3, our task is now to prove the convergence of the sequence of functionalsEh given by (4.10) towards
E[c] := Z Ω ∇p[c]·(rI+c)∇p[c] +ν γ (|r+c1| γ+|r+c 2|γ) dx, (4.11)
where p[c] ∈ H1(Ω) is a weak solution of the Poisson equation (4.3) subject to no-flux boundary conditions, andc1,c2 are the diagonal entries ofc=
c1 0 0 c2
. Similarly as in Section 3.2 we choose to work in the space L∞+(Ω)2diag×2 of diagonal nonnegative tensors on Ω with essentially bounded entries, equipped with the norm topology of L2(Ω). Note that for c ∈ L∞+(Ω)2diag×2 the Poisson equation (4.3) has a solution p[c]∈H1(Ω), unique up to an additive constant, and E[c]<+∞. Lemma 5. For any sequence of nonnegative diagonal tensors (cN)N∈N ⊂ L
∞
+(Ω)2
×2
diag with entries uniformly bounded in Lγ(Ω) and converging entrywise to c ∈ Lγ
+(Ω)2
×2
diag in the norm topology of
L2(Ω)as h= 1/N →0, we have,
E[c]≤ lim inf h=1/N→0
Eh[cN], (4.12)
with Eh given by (4.10) and E defined in (4.11).
Proof. Due to the strong convergence of the entries of cN in L2(Ω) there exist a subsequence converging almost everywhere in Ω to c. Then, we have by the Fatou Lemma,
Z Ω |r+c1|γdx≤ lim inf h=1/N→0 Z Ω |r+cN1 |γdx, (4.13)
which is finite due to the uniform boundedness of cN1 inLγ(Ω). Similarly forcN2 .
For the sequel let us denote p := p[c] ∈ H1(Ω) is a solution of the Poisson equation (4.3) with conductivity c, pN := p[cN] a solution of the Poisson equation (4.3) with conductivity cN
and ph := ph[cN] ∈ Wh a solution of the finite element discretization (4.4) with h = 1/N and conductivity cN. Then, by an obvious modification of the auxiliary Lemma 1 for diagonal tensor-valued conductivities we have by (2.3),
Z Ω ∇p·(rI+c)∇pdx= lim N→∞ Z Ω ∇pN ·(rI+cN)∇pNdx. (4.14)
Let us define the bilinear formsBN :H1(Ω)×H1(Ω)→R,
BN(u, v) =
Z
Ω
Note that BN(u, v) < +∞ for u, v ∈ H1(Ω) since cN ∈ L∞+(Ω)2diag×2. Moreover, since rI+cN is symmetric and positive definite, BN induces a seminorm onH1(Ω),
|u|BN := q
BN(u, u) foru∈H1(Ω). With this notation we have
Z Ω ∇pN ·(rI+cN)∇pNdx= pN 2 BN.
We now proceed along the lines of standard theory of the finite element method (proof of C´ea´s Lemma in the energy norm, see, e.g., [5]). Due to the Galerkin orthogonality
BN(pN −ph, ψ) = 0 for all ψ∈Wh,
(4.15)
we have, noting that ph ∈Wh,
pN 2 BN = p N−ph 2 BN+ p h 2 BN.
Then, again by (4.15) and by the Cauchy-Schwartz inequality, we have for all ψ∈Wh,
p N −ph 2 BN =B N(pN −ph, pN −ψ)≤ p N −ph BN pN −ψ BN.
Therefore, with the triangle inequality,
p N−ph BN ≤ψ∈infWh pN −ψ BN ≤ pN −p BN + inf ψ∈Wh|p−ψ|BN.
Due to the strong convergence ofcN →cinL2(Ω) and the standard result of approximation theory, see, e.g., [5], we have
lim h=1/N→0 ψinf∈Wh|p−ψ| 2 BN ≤ lim h→0ψ∈infWh Z Ω ∇(p−ψ)·(rI+c)∇(p−ψ) dx + lim N→∞ Z Ω ∇p·(cN −c)∇pdx = 0.
Due to (4.14) and the weak convergence of pN * pin H1(Ω), lim N→∞ pN −p BN = 0. (4.16)
Thus, collecting the above results from (4.14) up to (4.16), we conclude that
Z Ω ∇p·(rI+c)∇pdx= lim N→∞ pN 2 BN = lim h=1/N→0 p h 2 BN =h=1lim/N→0 Z Ω ∇ph·(rI+cN)∇phdx, which together with (4.13) gives (4.12).
Remark 2. Note that if γ > 1 and with the assumption that the sequence (cN)N∈N converges
(entrywise) in the norm topology of Lγ(Ω), the statement of Lemma 5 can be strengthened to
E[c] = lim h=1/N→0
Eh[cN].
This follows directly from the fact that in this case we have for the metabolic term Z Ω |r+c1|γ+|r+c2|γdx= lim h=1/N→0 Z Ω |r+cN1 |γ+|r+cN2 |γdx.
Lemma 5 and Remark 2 trivially imply the Γ-convergence of the sequence of energy functionals Eh in the norm topology ofLγ(Ω) forγ >1:
Theorem 2. Let γ >1, S ∈L2(Ω)with R
ΩSdx= 0and S h
i be given by (4.9). Then the sequence
Eh given by (4.10) Γ-converges toE defined in (4.11)with respect to the norm topology of Lγ(Ω)on
the set L∞+(Ω)2diag×2. In particular:
• For any sequence (cN)N∈N⊂L
∞
+(Ω)2diag×2 converging entrywise to c∈L γ
+(Ω)2diag×2 in the norm topology of Lγ(Ω)as h= 1/N →0, we have
E[c]≤ lim inf h=1/N→0
Eh[cN].
• For any c∈L∞+(Ω)diag2×2 there exists a sequence (cN)N∈N ⊂L
∞
+(Ω)2diag×2 converging entrywise toc∈Lγ+(Ω)2diag×2 in the norm topology of Lγ(Ω)as h= 1/N →0, such that
E[c]≥ lim sup h=1/N→0
Eh[cN].
Proof. The lim inf-statement follows directly from Lemma 5. For the lim sup-statement it is suffi-cient to setcN :=cfor all N ∈Nand use Remark 2, which in fact leads to the stronger statement
E[c] = lim h=1/N→0
Eh[cN].
4.4. Introduction of diffusion and construction of continuum energy minimizers (γ >1). As in the one-dimensional case, we introduce a diffusive term into the discrete energy functionals, which shall provide compactness of the sequence of energy minimizers. We again construct a piecewise linear approximation of the discrete conductivities C, which, however, turns out to be technically quite involved in the two-dimensional situation.
We shall describe the process for the conductivities of the horizontal edges, and by a slight abuse of notation, we denoteCi+1/2,j the conductivity of the horizontal edge connecting the node (ih, jh) to ((i+ 1)h, jh) for i= 0, . . . , N−1, j = 0, . . . , N where h = 1/N. Moreover, we denoteMi+1/2,j the midpoint of this edge, i.e., Mi+1/2,j = ((i+ 1/2)h, jh). For a given vector of conductivities C, we construct the continuous function Qh1[C] on Ω, such that
Qh1[C](Mi+1/2,,j) =Ci+1/2,j, fori= 0, . . . , N−1, j = 0, . . . , N,
and Qh1[C] is linear on each triangle spanned by the nodes Mi−1/2,j, Mi+1/2,j, Mi−1/2,j+1 and on each triangle spanned by the nodes Mi+1/2,j, Mi+1/2,j+1, Mi−1/2,j+1, for i = 1, . . . , N −1, j = 0, . . . , N −1. Let us denote the union of such two triangles, i.e., the square spanned by the nodes Mi−1/2,j,Mi+1/2,j,Mi−1/2,j+1 andMi+1/2,j+1, by Wij. Then, a simple calculation reveals that
Z Wij |∇Qh1[C]|2dx = 1 2 (Ci−1/2,j−Ci+1/2,j)2+ (Ci+1/2,j−Ci+1/2,j+1)2 (4.17) +(Ci+1/2,j+1−Ci−1/2,j+1)2+ (Ci−1/2,j+1−Ci−1/2,j)2 .
On the “boundary stripe” (0, h/2)×(0,1) and (1−h/2,1)×(0,1) the function is defined to be constant in the x-direction, such that it is globally continuous on Ω, i.e.,
Qh1[C](x) := C1/2,j+1−C1/2,j h (x2−jh) +C1/2,j, forx= (x1, x2)∈(0, h/2)×(jh,(j+ 1)h), Qh1[C](x) := CN−1/2,j+1−CN−1/2,j h (x2−jh) +CN−1/2,j, forx= (x1, x2)∈(1−h/2,1)×(jh,(j+ 1)h)
forj= 0, . . . , N−1. Summing up (4.17) over all squaresWij and the boundary stripe, we arrive at Z Ω |∇Qh1[C]|2dx=Dx[C], (4.18) with Dx[C] := N−1 X i=0 N−1 X j=0 (Ci+1/2,j−Ci+1/2,j+1)2+ N−1 X i=1 N−1 X j=1 (Ci−1/2,j−Ci+1/2,j)2 +1 2 N−1 X i=1 (Ci−1/2,0−Ci+1/2,0)2+ (Ci−1/2,N−Ci+1/2,N)2 .
Performing the same procedure for the vertical edges, we obtain
Z
Ω
|∇Qh2[C]|2dx=Dy[C], (4.19)
with obvious definitions of Qh2[C] and Dy[C].
Consequently, we define the sequence of discrete energy functionals Ediffh ,
Ediffh [C] :=D2(Dx[C] +Dy[C]) +Eh[C], (4.20)
with D2>0 diffusion constant and Eh[C] defined in (4.1), coupled to the Kirchhoff law (4.2) with sources/sinks Sh
i given by (4.9). We then have:
Proposition 4. For any vector C = (Ci)i∈E of nonnegative entries, we have
Ediffh [C] =D2
Z
Ω
|∇Qh1[C]|2+|∇Qh2[C]|2dx+Eh[Qh0[C]],
with Ediffh defined in (4.20) and Eh given by (4.10) with the pressures ph being a solution of the
FEM-discretized Poisson equation (4.4)with c=QN0 [C]. We are now in shape to prove the main result of this section: Theorem 3. Let γ >1,S ∈L2(Ω)withR
ΩSdx= 0and S h
i be given by (4.9). Let(CN)N∈N⊂RN
be a sequence of global minimizers of the discrete energy functionals Ediffh given by (4.20) with
h= 1/N. Then the sequence of diagonal 2×2 matrices
cN := Qh1[CN] 0 0 Qh2[CN] converges weakly in H1(Ω)2×2 to c∈H1(Ω)2×2
+ as h= 1/N → 0, with c a global minimizer of the functional Ediff :H+1(Ω)2diag×2 →R,
Ediff[c] :=D2
Z
Ω
|∇c1|2+|∇c2|2dx+E[c],
where E[c] is given by (4.11). Proof. Let us observe that
Ediffh [CN]≤Ediffh [0] = h 2 2 X i∈V X ?∈{E,W,N,S} r Pei−Pei,? h !2 +ν γr γ,
where (Pei)i∈V is a solution of the Kirchhoff law (4.2) with conductivities C = 0 and sources/sinks
˜
ph(Xi), i∈V, of the solution ˜ph of the discretized Poisson equation (4.4) with conductivity tensor
c= 0. Moreover, due to formula (4.5) we have
h2 2 X i∈V X ?∈{E,W,N,S} r Pei−Pei,? h !2 =r Z Ω |∇p˜h|2dx,
and the uniform boundedness of∇p˜h inL2(Ω) implies a uniform bound on Ediffh [CN]. Since the sequence
D2 Z Ω |∇Qh1[CN]|2+|∇Qh2[CN]|2dx=D2 Dx[CN] +Dy[CN] ≤Ediffh [CN]
is uniformly bounded, there exist subsequences of Qh1[CN] and Qh2[CN] converging to some c1,
c2 ∈ H1(Ω) weakly in H1(Ω), and strongly in L2(Ω). It is easy to check that then also Qh0[CN] converges toc:=
c1 0 0 c2
strongly inL2(0,1)2×2. Clearly, we also haveQh0[CN]∈L∞+(Ω)2
×2 diag with entries uniformly bounded in Lγ(Ω). Consequently, by Lemma 5, we have
E[c]≤ lim inf h=1/N→0E
h[cN].
Moreover, due to the weak lower semicontinuity of the L2-norm, we have
Z Ω |∇c1|2+|∇c2|2dx≤ lim inf h=1/N→0 Z Ω |∇Qh1[CN]|2+|∇Qh2[CN]|2dx. Consequently,
Ediff[c]≤ lim inf h=1/N→0E
h
diff[CN]. (4.21)
We claim thatc is a global minimizer ofEdiff inH+1(Ω)2
×2
diag. For contradiction, assume that there existsc∈H+1(Ω)2diag×2 such that
Ediff[c]<Ediff[c].
We define the sequence (CN)N∈N by setting the conductivity C
N
i of each horizontal edge i∈E to the average of c1 over the two triangles Ti;1,Ti;2 ∈ Th that contain the edgei, i.e.,
CNi := 1
h2
Z
Ti;1∪Ti;2
c1(x) dx.
Similarly, we use the averages of c2 to define the conductivities of the vertical edges. Then, by assumption, we have for all h= 1/N,N ∈N,
Ediffh [CN]≥Ediffh [CN].
(4.22)
It is easy to check that the sequence Qh1[C N
] converges strongly in H1(Ω) towardsc1, therefore
Z Ω |∇Qh1[C N ]|2dx→ Z Ω |∇c1|2dx ash= 1/N →0, and analogously for Qh2[C
N
] and c2. Moreover, the sequence Qh0[C N
] converges to c strongly in
Lγ(Ω)2diag×2, therefore, by Remark 2, Eh[
QN0 [C N ]]→ E[c] ash= 1/N →0. Consequently, lim h=1/N→0E h diff[C N ] =Ediff[c]<Ediff[c], a contradiction to (4.21)–(4.22).
Remark 3. We can easily generalize to the situation when the two-dimensional grid is not rectan-gular, but consists of parallelograms with sides of equal length in linearly independent directions θ1,
θ2 ∈S1, where S1 is the unit circle in R2. Then the coordinate transform
(1,0)7→θ1, (0,1)7→θ2 in (4.11)leads to the transformed continuum energy functional
E[c] = Z Ω ∇p[c]·P[c]∇p[c] +ν γ (|r+c1| γ+|r+c 2|γ) dx coupled to the Poisson equation
−∇ ·(P[c]∇p) =S
with the permeability tensor
P[c] =rI+c1θ1⊗θ1+c2θ2⊗θ2. The eigenvalues of P[c] (principal permeabilities) are
λ1,2 = 1 2 c1+c2± p (c1−c2)2−4c1c2(θ1·θ2)2
and the corresponding eigenvectors (principal directions)
u1,2 =θ1+ c2−c1± p (c1−c2)2−4c1c2(θ1·θ2)2 2c1θ1·θ2 θ2. 5. Appendix
Here we provide more technical details for the constructions and calculations performed in Sec-tion 4.1.
5.1. Linear basis functions. We list the explicit definitions for the piecewise linear basis functions on the triangulation Th, constructed in Section 4.1. Any interior node i ∈
V has six adjacent
triangles, denoted clockwise by TiN E, TiSE,TiS, TiSW, TiN W,TiN, see Fig. 1. For each triangle we construct three basis functions, supported on the respective triangle and linear on their support. Obviously, the basis functions are uniquely determined by their values on the triangle vertices. For later reference we list their gradients, which are constant vectors on the respective triangles.
• On the N E-triangle TiN E we construct the linear basis functions φN Ei;1 , φN Ei;2 , φN Ei;3 defined by φN Ei;1 (Xi) = 1, φN Ei;1 (Xi,E) = 0, φN Ei;1 (Xi,N) = 0, φN Ei;2 (Xi) = 0, φN Ei;2 (Xi,E) = 1, φN Ei;2 (Xi,N) = 0, φN Ei;3 (Xi) = 0, φN Ei;3 (Xi,E) = 0, φN Ei;3 (Xi,N) = 1, so that ∇φN Ei;1 ≡ −1 h(1,1), ∇φ N E i;2 ≡ 1 h(1,0), ∇φ N E i;3 ≡ 1 h(0,1), on T N E i .
• On theSE-triangle TiSE we construct the linear basis functions φSEi;1,φSEi;2,φSEi;3 defined by φSEi;1(Xi) = 1, φSEi;1(Xi,E) = 0, φSEi;1(Xi,SE) = 0, φSEi;2(Xi) = 0, φSEi;2(Xi,E) = 1, φSEi;2(Xi,SE) = 0, φSEi;3(Xi) = 0, φSEi;3(Xi,E) = 0, φSEi;3(Xi,SE) = 1, so that ∇φSEi;1 ≡ −1 h(1,0), ∇φ SE i;2 ≡ 1 h(1,1), ∇φ SE i;3 ≡ − 1 h(0,1), on T SE i .
• On theS-triangle TiS we construct the linear basis functions φSi;1,φSi;2,φSi;3 defined by
φSi;1(Xi) = 1, φSi;1(Xi,SE) = 0, φSi;1(Xi,S) = 0, φSi;2(Xi) = 0, φSi;2(Xi,SE) = 1, φSi;2(Xi,S) = 0, φSi;3(Xi) = 0, φSi;3(Xi,SE) = 0, φSi;3(Xi,S) = 1, so that ∇φSi;1≡ 1 h(0,1), ∇φ S i;2≡ 1 h(1,0), ∇φ S i;3≡ − 1 h(1,1), onT S i .
• On the SW-triangle TiSW we construct the linear basis functions φSWi;1 , φSWi;2 ,φSWi;3 defined by φSWi;1 (Xi) = 1, φSWi;1 (Xi,S) = 0, φSWi;1 (Xi,W) = 0, φSWi;2 (Xi) = 0, φSWi;2 (Xi,S) = 1, φSWi;2 (Xi,W) = 0, φSWi;3 (Xi) = 0, φSWi;3 (Xi,S) = 0, φSWi;3 (Xi,W) = 1, so that ∇φSWi;1 ≡ 1 h(1,1), ∇φ SW i;2 ≡ − 1 h(0,1), ∇φ SW i;3 ≡ − 1 h(1,0), on T SW i .
• On theN W-triangleTiN W we construct the linear basis functionsφN Wi;1 ,φN Wi;2 ,φN Wi;3 defined by φN Wi;1 (Xi) = 1, φN Wi;1 (Xi,W) = 0, φN Wi;1 (Xi,N W) = 0, φN Wi;2 (Xi) = 0, φN Wi;2 (Xi,W) = 1, φN Wi;2 (Xi,N W) = 0, φN Wi;3 (Xi) = 0, φN Wi;3 (Xi,W) = 0, φN Wi;3 (Xi,N W) = 1, so that ∇φN Wi;1 ≡ 1 h(1,0), ∇φ N W i;2 ≡ − 1 h(1,1), ∇φ N W i;3 ≡ 1 h(0,1), onT N W i .
• On theN-triangle TiN we construct the linear basis functions φNi;1,φNi;2,φNi;3 defined by
φNi;1(Xi) = 1, φNi;1(Xi,N W) = 0, φNi;1(Xi,N) = 0, φNi;2(Xi) = 0, φNi;2(Xi,N W) = 1, φNi;2(Xi,N) = 0, φNi;3(Xi) = 0, φNi;3(Xi,N W) = 0, φNi;3(Xi,N) = 1, so that ∇φNi;1≡ −1 h(0,1), ∇φ N i;2 ≡ − 1 h(1,0), ∇φ N i;3≡ 1 h(1,1), on T N i .
5.2. Gradients of ph. Here we provide the gradient of the solution ph ∈ Wh of (4.4), constructed in Section 4.1. Sinceph is continuous on Ω and linear on each triangle in Th, it is represented by its vertex valuesPih :=ph(Xi),i∈V. Then, for any interior nodei∈V
we readily have ∇ph= 1 h (Ph i,E−Pih, Pi,Nh −Pih) onTiN E, (Pi,Eh −Pih, Pih−Pi,SEh ) onTiSE, (Ph i,SE−Pih, Pih−Pi,Sh ) on TiS, (Pih−Pi,Wh , Pih−Pi,Sh ) onTiSW, (Pih−Pi,Wh , Pi,N Wh −Pih) on TiN W, (Pih−Pi,N Wh , Pi,Nh −Pih) on TiN.
5.3. Explicit calculation for (4.8). Finally, we provide the detailed calculation for the identity (4.8). Noting thatψih is supported onUi =TiN E∪TiSE∪TiS∪TiSW ∪TiN W ∪TiN, and taking into account the results listed in Sections 5.1 and 5.2, we have
Z TN E i ∇ph·rI+Qh0[C] ∇ψhi dx = r+C E i 2 Pih−Pi,Eh +r+C N i 2 Pih−Pi,Nh , Z TSE i ∇ph·rI+ Qh0[C] ∇ψh i dx = r+CiE 2 Pih−Pi,Eh , Z TiS ∇ph·rI+Qh0[C] ∇ψhi dx = r+C S i 2 Pih−Pi,Sh , Z TSW i ∇ph·rI+Qh0[C] ∇ψhi dx = r+C W i 2 Pih−Pi,Wh +r+C S i 2 Pih−Pi,Sh , Z TN W i ∇ph·rI+ Qh0[C] ∇ψh i dx = r+CiW 2 Pih−Pi,Wh , Z TiN ∇ph·rI+Qh0[C] ∇ψhi dx = r+C N i 2 Pih−Pi,Nh .
Summing up, we arrive at
Z Ω ∇ph·(rI+Qh0[C])∇ψihdx = (r+CiE) Pih−Pi,Eh + (r+CiN)Pih−Pi,Nh + (r+CiW)Pih−Pi,Wh + (r+CiS)Pih−Pi,Sh , which is (4.8). Acknowledgments
LMK was supported by the UK Engineering and Physical Sciences Research Council (EP-SRC) grant EP/L016516/1 and the German National Academic Foundation (Studienstiftung des Deutschen Volkes).
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